# Circle A has a radius of 2  and a center of (5 ,7 ). Circle B has a radius of 4  and a center of (3 ,2 ). If circle B is translated by <2 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Oct 25, 2017

$\text{circles touch externally}$

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare }}$ the distance (d) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of the radii}}$

• " if sum of radii">d" then circles overlap"

• " if sum of radii" < d" then no overlap"

We require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

$\text{under a translation } < 2 , - 1 >$

$\left(3 , 2\right) \to \left(3 + 2 , 2 - 1\right) \to \left(5 , 1\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

Note that the x-coordinate of the centres of both circles is 5 indicating that they lie on the vertical line x = 5
Hence d is the difference in the y-coordinates.

$\Rightarrow d = 7 - 1 = 6$

$\text{sum of radii } = 2 + 4 = 6$

$\text{since sum of radii } = d = 6$

$\text{then the circles touch externally}$
graph{((x-5)^2+(y-7)^2-4)((x-5)^2+(y-1)^2-16)=0 [-20, 20, -10, 10]}